Grundy domination number has attracted considerable attention [1]}, [2]}, [3]}, [4]}, and so had its total version, which was introduced in [5]} and studied further in [1]}, [7]}, [8]}, [3]}, [10]}. The concept was motivated by domination games, and was surveyed within the recent book [11]}.
Brešar et al. found a close connection between the Grundy domination number and the zero forcing number of a graph [12]}, which motivated the introduction of the Z-Grundy domination number \(\gamma _{gr}^{Z}(G)\)  of a graph \(G\) , which is just the dual of the zero forcing number \(Z(G)\) . Namely, in any graph the complement of any (minimum) zero forcing set is a (maximum) Z-set and vice versa. In particular, in every graph \(G\) , \(\gamma _{gr}^{Z}(G)=n(G)-Z(G)\) . Lin in [13]} further explored the relations between all four types of Grundy domination numbers and some concepts from linear algebra.
